CiteSeerX — Coding and Modulation for Code-Division Multiple Accessing

Coding and Modulation for Code-Division Multiple Accessing ^*

James L. Massey

Signal and Information Processing Laboratory Swiss Federal Institute of Technology

CH-8092 Zürich

Abstract: It is argued that the "rightful purpose" of a CDMA modulation system is to create a good channel for the coding system. This viewpoint leads to a precise formulation of energy efficiency and a novel formulation of bandwidth occupancy as normalized Shannon bandwidth. Various modulation/coding architectures are formulated and compared. It is shown that it is possible to design systems in such a way that the burden of separating and detecting the users' information sequences is divided between the demodulator and the decoders with resulting reduced complexity but with no penalty in either energy efficiency or bandwidth occupancy.

1 Introduction

Before one begins to study modulation systems and coding systems in detail for code-division multiple accessing (or for any other kind of

communications for that matter), it is well to reflect on what the purposes of these two systems are. As self-evident as this admonition may seem, we must remark that our experience shows that it is heeded more in the breech than in the practice. The reader is thus asked to bear with a recitation of these purposes that is not at all new but that is prone to be forgotten.

For our purposes, the model of an "L-chip synchronous CDMA channel"

shown in Fig. 1 will be adequate. As shown, the model describes one use of a memoryless channel. Each of the M-users, say user i, sends a symbol B_i to his corresponding modulator. We assume that B_i is a random variable with second

*The research reported here was supported by the European Space Research and Technology Centre (ESTEC) in Noordwijk, The Netherlands, under ESTEC Contract No. 8696/89/NL/US

Modulator 1

Modulator 2

Modulator K

. . .

Gaussian Noise Source

Demodulator . . .

R1

R2

RK

Y^L Z^L B1

B2

BK

B1

s

B2

s

BK

s

Fig. 1: Model of an L-chip synchronous CDMA channel.

( E[B_i2] ≤ E_c, s_iT s_i = L, and Z^L is white Gaussian noise of variance N_o/2.)

moment E[B_i2] ≤ E_c. We further assume that the random variables B₁, B₂, ..., B_M are independent, which is the essential difference between a multiple-access channel and a single-user channel. Using the usual signal-space representation with respect to any convenient basis of orthonormal waveforms, we can view the modulator as converting B_i into the signal vector B_is_i, where s_i = (s_i1, s_i2, ..., s_iL) is an L-component vector with energy L. For convenience, we will consider the components of s_i to be real numbers, i.e., s_i

∈

considering a baseband channel (but all our conclusions carry directly over to the passband case when these components are complex numbers). The energy of s_i is thus

s_iTs_i =

∑

n=1 L

s_in2 = L (1)

(where the superscript T denotes transposition) and our normalization of this energy to L is motivated by the usual case in CDMA systems where s_in

∈

for all n. The demodulator input is the received vector

Y^L =

∑

i=1 M

B_i s_i + Z^L (2)

where Z^L = (Z_1, Z₂, ..., Z_N) is white Gaussian noise (WGN), i.e., its components are independent zero-mean Gaussian random variables all with the same variance, which as usual we take to be N_o/2. We will be less explicit about the nature of the demodulator outputs, since these are at the choice of the designer of the modulation system.

We can now address the question of what the modulation system of Fig. 1 should be designed to do. But this requires us to think more carefully about how the users will actually send information over this channel. In fact, we know that, for reliable and efficient transmission, each user will encode his information bits into a sequence of symbols to be sent over the channel, and the receiver will decode the corresponding received sequence provided by the demodulator to recover these information bits. The conclusion is inescapable:

(i) The rightful purpose of the modulation system in Fig. 1 (i.e., of the K modulators and the demodulator) is to

create a good channel for coding.

To make this general purpose more directly applicable, we must separately consider the modulators and demodulator. The choice of the modulators, i.e., the choice of the sequences s₁, s₂, ..., s_L, determines a multiple-access channel with input B = (B₁, B₂, ..., B_M) and output Y^L = (Y₁, Y₂, ..., Y_L). The fundamental measure of goodness of a channel for coding is, of course, its capacity

(appropriately defined as we will later do) so we must conclude:

(ii) The rightful purpose of the K modulators in Fig. 1 is to create a channel from B to Y^L with maximum capacity.

The question of what the demodulator should do is more subtle. If we continue to take capacity as our only criterion of goodness, the answer is simple:

do nothing! Any processing of Y^L can only reduce capacity since, by the Data Processing Theorem [1, p. 158] of information theory, I(B; Y^L) ≥ I(B; R) where R = (R₁, R₂, ..., R_M) and where I(. ; .) denotes mutual information. Thus, taking R_i = Y^L for all i (so that we really have only one demodulator output) is optimum in the sense of maximizing capacity. But a channel "good for coding" should mean not only a channel with large capacity but also one that is convenient for coding and decoding. Because (at least roughly) the complexity of decoding grows with the number of quantization levels at the demodulator output, we can conclude the following:

(iii) The rightful purpose of the demodulator in Fig. 1 is to quantize the received vector Y^L as coarsely as possible consistent with a tolerably slight decrease of capacity.

The three conclusions stated above are, as we have already said, not new.

They appear quite explicitly in [2, p. 211], [3], [4], [5] and [6], sometimes with cut-off rate (which is also an excellent measure of channel quality for coding in single- user channels) used in place of capacity, but this makes no essential difference. In the following sections, we will study modulation and coding for the CDMA channel in light of these rightful purposes of the modulation system. The

novelty of much of what follows suggests that although these "rightful purposes"

are virtually self-evident, they are usually ignored in the design and analysis of CDMA systems.

The two aspects of the modulation system of Fig. 1 that are the most frequent source of controversy are its energy efficiency and its "bandwidth

efficiency". We now wish to define these terms in such a way that we can validly compare systems with different values of L and E_c, but of course we would fix K (the number of users) and N_o (the one-sided noise power spectral density) in any such comparison. Energy efficiency is the more easily understood of these two

"efficiencies" so we begin with it.

The components s_in of the "spreading sequence" s_i are traditionally called chips. Since user i sends L chips with total energy E [B_i s_iT s_i B_i] = L E [B_i2] ≤ L E_c , where we have made use of (1), we see that E_c is the (maximum allowed) average chip energy for user i. We may draw the immediate conclusion:

(iv) When comparing modulation systems of the type in Fig. 1 with the same values of K and E_c, the appropriate measure of energy utilization is the ratio of the per-user capacity per chip of the channel from B to Y^L to the chip energy E_c.

By the chip time T_c of a CDMA system, we mean the time interval

allocated to the transmission of one chip over the underlying waveform channel.

More precisely, the chip rate 1/T_c is the (average) number of chips per second produced by a modulator. The chip rate 1/T_c is precisely the Shannon bandwidth of the modulated signals, i.e., the number of dimensions of signal space used per second, cf. [6]. This Shannon bandwidth is exactly twice the ordinary Fourier

bandwidth when the corresponding chip waveforms are chosen to minimize the Fourier bandwidth. If two modulation systems (possibly with different values of E_c and L) are such that the first has three times the capacity per chip of the second, then (upon the assumption that the two corresponding coding systems operate at the same fraction of capacity) to accomodate the same information transfer rate one will have to send three times as many chips per second for the second system as for the first, i.e., the second system requires three times as much bandwidth. In fact, the reciprocal of the total capacity per chip, i.e., the numbers of chips required per bit of total capacity, can and will be defined here as the normalized Shannon bandwidth. We need merely to multiply this normalized bandwidth by the desired total information bit rate to obtain the Shannon bandwidth; a further multiplication by one-half yields the Fourier bandwidth (or a close

approximation thereto). We are forced to the following conclusion.

(v) When comparing modulation systems of the type in Fig. 1 with the same values of K and E_c, the appropriate measure of bandwidth occupancy is normalized Shannon bandwidth defined as the

reciprocal of the total capacity per chip of the channel from B to Y^L.

It should now be apparent that the demands for energy efficiency and bandwidth "efficiency" are to some extent conflicting. Because the capacity per chip generally (but not always as we will see in Section 4 below) increases without limit as the chip energy E_c increases, one can usually be as bandwidth "efficient"

as one pleases -- if one doesn't care about energy efficiency. "Bandwidth efficiency" is misleading terminology that should be expunged from the vocabulary of communications engineers; instead, one should speak of

bandwidth occupancy. There is, however, an upper limit on energy utilization (as we will see below) so that it does make good sense to talk about energy efficiency.

2 Trivial Demodulation with Joint Decoding

As noted above, the optimum demodulator, in the sense of maximizing the capacity of the channel from B to R in Fig. 1, is that which does nothing (expect of course to project the received waveform into its representation Y^L in signal space. Thus, it is natural to begin our detailed modulation/coding studies with the architecture of Fig. 2 in which the demodulator merely passes Y^L along to the joint decoder that has full responsibility for separating and detecting the

sequences of information bits from the K information sources. We now consider the appropriate notion of capacity for the channel from B to Y^L, which will serve as a useful upper bound on the capacity from B to R for less trivial demodulators.

Modulator 1

Modulator 2

Modulator K

. . .

Gaussian Noise Source

Joint Decoder . . . Y^L

Z^L B1

B2

BK

B1

s

B2

s

BK

s

Y^L

Trivial Demodulator

Fig. 2: Trivial demodulation combined with joint decoding--the joint decoders emits decision on the information bits of the K users.

It would seem at first glance that the appropriate notion of capacity for the multiple-access channel would be the per-user sum capacity per chip, C_sum, from B to Y^L, i.e., the maximum of 1

KL I (B; Y^L), where the maximum is taken over all choices of probability distributions for the independent random variables B₁, B₂, ..., B_K subject to the constraint that E [B_i2] ≤ E_c for all i. It is easy to obtain a simple upper bound on this capacity. The total average modulated signal energy per chip is at most KE_c with equality if and only if E [B_i2] ≤ E_c for all i. The additive Gaussian noise on each chip has variance N_o/2. Thus, 1

KL I (B; Y^L) is upper bounded by 1

K times the capacity of a single-user discrete-time Gaussian channel with signal energy KE_c and noise variance N_o/2. By one of the oldest results in information theory (cf. [2, p. 147[) this capacity is 1

2 log²(1 + KE_c

N_o/2 ) with equality if and only if Y^L is WGN with variance KE_c + N_o/2.

Proposition 1: The per-user sum capacity per chip of the channel from B to Y^L satisfies

C_sum ≤ 1

2K log²(1 + KE_c

N_o/2 ) [bits/chip] (3)

with equality if and only if Y^L is WGN with variance KE_c + N_o/2.

But in fact the sum capacity is not the capacity of most interest for multiple-accessing since it places no demands on fairness. What is of much greater interest is the per-user symmetric capacity per chip, C_sym, which is defined as the maximum number C such that the K-tuple (C, C, .., C) lies in the capacity region of the channel, i.e., such that all of the K users can simultaneously send information reliably at any per-user rate less than C bits/chip. The following somewhat surprising result is proved in [7].

Proposition 2: The per-user symmetric capacity per chip of the channel from B to Y^L satisfies

C_sym ≤ 1

2K log²(1 + KE_c

N_o/2 ) [bits/chip] (4)

with equality in the (usual) case where K ≥ L if and only if the L × K matrix S with columns s₁, s₂, ... , s_K has mutually orthogonal and equal-energy rows.

We will call sequences s₁, s₂, ... , s_K satisfying the conditions for equality in Proposition 2 a Welch-Bound-Equality (WBE) sequence set (more precise

terminology would be "sequence multiset" as the sequences s₁, s₂, ... , s_K need not be distinct), since these are precisely the conditions that were proved in [8] for equality to hold in a lower bound due to Welch [9] on the sum of the "squared correlations" (s_iT s_j)² over all i and j. Because C_sym ≤ C_sum, we see from (4) and (6) that fairness can be achieved with no loss of capacity by choosing the spreading sequences s₁, s₂, ... , s_K to form a WBE sequence set.

Because log₂(1 + KE_c

N_o/2 ) ≤ 1.44 KE_c

N_o/2 with equality when and only when KE_c

N_o/2 is sufficiently small (say, KE_c

N_o/2 ≤ 1), we obtain the following from (4).

Proposition 3: The per-user symmetric capacity per chip of the channel from B to Y^L satisfies

C_sym ≤ 1.44 E_c

N_o (5)

with near equality in the case K ≥ L when and only when E_c

N_o is sufficiently small (say, E_c

N_o ≤ 1

2K ) and the spreading sequences form a WBE sequence set.

It is now natural [and in keeping with conclusion (iv) of Section 1] to define the energy efficiency γ of a CDMA modulation system by

γ = C_sym/(1.44 E_c

N_o) (6)

since this percentage cannot exceed 1 but approaches 1 when E_c N_o ≤ 1

2K and a WBE sequence set is used.

3 Full Demodulation with Single-User Decoding

The main drawback of the modulation/coding architecture of Fig. 2 is that it places enormous demands on the "joint decoder." Decoding on a single-user channel is generally complex enough; joint decoding for a K-user channel with, say, K ≈ 10⁴ can be mind-boggling in complexity. For this reason, most past

CDMA systems use the architecture of Fig. 3, in which by a single-user decoder for user i is meant a decoder that treats the contributions of all other users to its input as constituting additional white Gaussian noise. The demodulator in Fig. 2 has been given full responsibility for separating the signals of the K users!

Modulator 1

Modulator 2

Modulator K

. . .

Gaussian Noise Source

. . . Y^L

Z^L B1

B2

BK

B1

s

B2

s

BK

s

Full Demodulator

Decoder 2

Decoder K Decoder 1 R1

R2

RK

Fig. 3: Full demodulation with single-user decoding wherein the decoder for user i treats the effects of all other users on R_i as white Gaussian noise.

Consider without loss of essential generality the situation for user 1, and assume here and hereafter that E[B_i] = 0 and that E[B_i2] =E_c for all i, which

conditions entail no loss of optimality and are generally desirable in practice. The optimum demodulator will, of course, supply to the single-user decoder for user 1 the number R₁ determined by correlating Y^L with user 1's spreading sequence s₁, i.e.,

R₁ = 1

√

√

where

N₁ = 1

√

is a zero-mean Gaussian random variable with variance N_o/2 and where

W₁ = 1

√

K

B_j s_1T s_j (8)

is a zero-mean random variable with variance

E_c L ∑

j=2 K

(s_1T s_j)².

By the central-limit theorem (cf. [10, p. 256]), one sees from (8) that W₁ will be well approximated by a Gaussian random variable for a WBE sequence set and, indeed, most other sequence sets of practical interest. But Welch's bound [9]

effectively provides a lower bound on the sum of the variances of the random variables W₁, W₂, ... , W_K that was shown in [8] to hold with equality if and only if the sequence set is WBE. It was further proved in [8] that, for a WBE sequence set, these variances all have the same value, namely E_c(K - L). Thus, we see that (7) describes a single-user Gaussian channel from B_i to R_i with signal energy L Ec and noise variance at least E_c(K - L) + N_o/2 where the minimum holds if and only if the sequence set is WBE. Noting that user i uses this equivalent channel only once every L chips, we may summarize our observations as follows.

Proposition 4: The per-user single-user-decoding capacity per chip of the channel from B to R for the (usual) case K ≥ L satisfies

C_sud ≤ 1

2L log²[1 + L E_c

N_o/2 + E_c(K - L) ] [bits/chip] (9)

with equality if and only if s₁, s₂, ... , s_K form a WBE sequence set. Moreover, if K ≥ 2L or if N_o/2 ≥ E_c(K - L), then C_sud is well approximated for a WBE sequence set by

C_sud ≈ 0.72 E_c

N_o/2 + E_c(K - L) [bits/chip]. (10)

The first of the two conditions for the validity of the approximation (10) follows from the fact that if K ≥ 2L, then the second term in square brackets in (9) is less than 1 so that the logarithm is well approximated by log₂(e) = 1.44 times that term.

We see from (9) that the approximation

C_sud ≈ 1.44 E_c

N_o [bits/chip] (11)

holds when and only when the sequence set is WBE and, moreover, E_c(K - L) is

sufficiently small compared to N_o/2, say E_c

N_o ≤ 1

4(K - L) . But (11) is precisely the condition for near 100% energy efficiency, i.e., for γ = 1 in (6). Note also that the condition E_c

N_o ≤ 1

4(K - L) is only about twice as restrictive (since K ≥ 2L is assumed and usually K >> L) as the condition E_c

N_o ≤ 1

2K required for near 100%

energy efficiency with joint decoding. We can conclude the following.

Both a well-designed single-user-decoding system and a joint-decoding system can operate as close to 100% energy efficiency as desired by choosing a suitably small "signal-to-noise ratio" E_c

N_o . Moreover, the upper limit of

"signal-to-noise ratios" E_c

N_o for which near 100% energy efficiency is possible is only about twice as large for joint decoding as for single-user decoding.

4 Bandwidth Comparisons

As we have just seen, suboptimum (but simple) single-user decoding implies no unavoidable penalty in energy efficiency compared to optimum (but complex) joint decoding. We now show, however, that single-user decoding does entail a substantial bandwidth penalty compared to joint decoding.

Consider first single-user decoding. To operate at energy efficiency γ requires, by the definition (6), that

C_sud = 1.44 γ E_c

N_o [bit/chip]. (12)

Equating the right sides of (10) and (12), then solving for the "signal-to-noise ratio" E_c

N_o yields

E_c

N_o = 1 - γ

γ (1 - L/K) 2 K .

Substituting this into (12) gives the following expression for the normalized Shannon bandwidth that, like (10), is a close approximation whenever K ≥ 2L.

(K C_sud)^-1 ≈ (1 - L/K) 1.39

1 - γ [chips/bit]. (13)

The situation for joint decoding is somewhat more complicated. Again, we begin with

C_sym = 1.44 γ E_c

N_o [bits/chip] (14)

and then equate the right sides of (4) and (10) to obtain the following transcendental equation for the corresponding "signal to-noise" ratio:

2 γ K E_c

N_o = ln(1 + 2KE_c

N_o). (15)

When E_c is sufficiently small (which implies γ ≈ 1), one finds from (15) that

E_c

N_o ≈ 1 - γ K ,

which when used with (14) yields the following approximation for the normalized Shannon bandwidth:

(K C_sym)^-1 ≈ 0.69

1 - γ , (16)

which holds when K ≥ 2L and γ ≈ 1. Comparing (13) and (16) shows that, in the usual case when K >> L,

the bandwidth required to operate at a given high energy efficiency γ is about twice as great for a single-user-decoding system as for a joint- decoding system.

But high energy efficiency is in fact the most favorable case for the bandwidth required with single-user decoding compared to that for joint

decoding, as one sees from Table I where we have listed the required normalized Shannon bandwidths for single-user decoding [obtained from (13)] and for joint decoding [obtained by solving the transcendental equation (15)]. At an energy efficiency of 20%, which is in the region where one could expect a practical

CDMA system to operate, one sees that a single-user-decoding system will require

well more than three times the bandwidth of a joint-decoding system in the usual case where K >> L. One is tempted to speculate that the poor "spectral efficiencies" that have sometimes been reported for CDMA systems are rather the result of their designers' predilection for single-user processing rather than of any inherent spectral disadvantage of CDMA systems compared to other multiple- access systems (there is no such disadvantage!).

γ (K C_sud)^-1 (K C_sym)^-1

[chips/bit] [chips/bit]

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

1 ∞ ∞

.9 13.9 (1 - L/K) 6.71

.8 6.95 (1 - L/K) 3.23

.7 4.63 (1 - L/K) 2.06

.6 3.48 (1 - L/K) 1.47

.5 2.78 (1 - L/K) 1.11

.4 2.32 (1 - L/K) 0.86

.3 1.99 (1 - L/K) 0.67

.2 1.74 (1 - L/K) 0.52

.1 1.54 (1 - L/K) 0.38

.01 1.40 (1 - L/K) 0.21

.001 1.39 (1 - L/K) 0.15

≈ 0 1.39 (1 - L/K) ≈ 0

Table I: Normalized Shannon bandwidths (K C_sud)^-1 and (K C_sym)^-1 for single-user decoding (with K ≥ 2L assumed) and joint-decoding CDMA systems, respectively, as a function of energy efficiency γ.

One sees also from Table I that one cannot reduce the normalized

Shannon bandwidth of the single-user-decoding system below 1.39 (1 - L/K), no matter how inefficiently one is willing to use energy. This is a result of the fact that, as follows from (9),

KC_sud → K 2L log²

1

1 - L/K [bits/chip] as E^c → ∞ ,

which implies when K >> L that

KC_sud ≤ 0.72 [bits/chip]

with near equality for large E_c. In other words, when K >> L, no amount of chip energy can give a total capacity of more than 0.72 [bits/chip] if single-user

decoding is used. Hence, the normalized Shannon bandwidth cannot be reduced below 1/0.72 = 1.39 [chips/bit]. For joint decoding, however, one sees from (4) that

KC_sym → ∞ as E_c → ∞ ,

which is why the normalized Shannon bandwidth can be made as small as desired if one is will to live with small energy efficiencies.

5 Partial Demodulation with User-Group Decoding

As we will now show, it is possible to make a compromise between single- user decoding and joint decoding that combines the best features of both, namely, the reduced complexity of the former and the reduced bandwidth of the latter.

[The ideas in this section represent joint work of the author and his colleague, Dr.

Thomas Mittelholzer.] The "hybrid" architecture is shown in Fig. 4.

Modulator 1

Modulator 2

Modulator K

. . .

Gaussian Noise Source

. . . Y^L

Z^L B1

B2

BK

B1

s

B2

s

BK

s

Partial Demodulator

R1

R2

RL

Decoder for user group 1

Decoder for user group 2

Decoder for user group L Fig. 4: Partial demodulation into signals for L groups of m users (K = mL)

followed by decoding for each group of users.

We assume that K = mL for some positive integer m and that the users are

partitioned into L groups of m users each. We further assume that s₁, s₂, ... , s_L are mutually orthogonal sequences (i.e., s_iT s_j = 0 for i ≠ j), each of energy L, and that all users in the i-th user group G_i use the same spreading sequence s_i. The L × K = L × mL sequence matrix S described in Proposition 2 can then be taken as

S = [M : M : ... : M] (17)

where M is the matrix whose columns are s₁, s₂, ... , s_L. By hypothesis,

M^T M = L I_L

where I_L is the L × L identity matrix. But M^T is then (within a constant factor) the inverse of M so that

M M^T = L I_L (18)

also holds, i.e., the rows of M are also mutually orthogonal and each of energy L.

It follows now from (17) that the rows of S are also mutually orthogonal and each of energy mL = K. Thus, the sequence set for the K = mL users is in fact a WBE sequence set and hence, by Proposition 2, delivers a per-user symmetric capacity for the channel from B to Y^L satisfying

C_sym = 1

2K log²(1 + KE_c

N_o/2 ) [bits/chip], (19)

which is the best possible.

Now, suppose that the "partial demodulator" of Fig. 4 correlates each of the L sequences assigned to the user groups with the received sequence Y^L (which requires only L = K/m correlations compared to K correlations for the full

demodulator considered in Section 2.) This partial demodulator then delivers to the decoder for the i-th user group the output

R_i = 1

√

√

∑

j ∈ G_j

B_j + N_i (20)

where we have used (2) and the mutual orthogonality of s₁, s₂, ... , s_L and where

N_i = 1

√

is a zero-mean Gaussian random variable of variance N_o/2. We can recover Y^L from R = (R₁, R₂, ... , R_L) in the manner

j=1∑ L

∑ i=1

L 1

√

∑ i=1

L 1

L s_j s_iT Y^L (21)

= 1

L M M^T Y^L = Y^L

where we have made use of (18). This means that the capacity of the channel from B to Y^L is exactly the same as that from B to R. Moreover, we note that the Gaussian random variables N₁, N₂, ... , N_L are independent. It follows then from (20) that the inputs Bj for j ∈ G_i are independent of all demodulator outputs except R_i. Hence, the per-user symmetric capacity C_i of the channel from these inputs to R_i must be precisely the same as the per-user symmetric capacity from B to R, i.e.,

C_i = 1

2K log²(1 + KE_c

N_o/2 ) [bits/chip] (22)

so that no capacity loss has resulted from our partial demodulation. The decoder for the i-th user group can now operate on the sequence of symbols from the single demodulator output labelled R_i to separate and detect the users only in this group and without having to worry at all about the other users. Absolutely no penalty in either energy efficiency or bandwidth occupancy compared to optimum joint decoding is incurred!

Many variations on the above hybrid scheme are possible, but the practical advantage of dividing the task of separating and detecting all M users between the modulation system and the coding system should be sufficiently clear to the reader from the simple scheme that we have described above

6 Concluding Remarks

The reader will certainly have noticed that in our analyses above we have suppressed many important practical aspects of CDMA systems such as lack of synchronization, different user energies, time variations, multipath effects, and the like. This was not because including these details would have altered our conclusions (they would not!) but rather because their inclusion would obscure the conceptual lessons that we were determined to set forth as nakedly as

possible. We do not mean to suggest that treating these details is easy. However, unless we keep our eyes clearly fixed on the "rightful purposes" of a CDMA system and on the basic information theory of such systems, we are only too apt to lose sight of the forest because of the many trees of detail.

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